Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;

int main() {
	// your code goes here
	setprecision(30);
	int n,i,j;
	cin>>n;
	double p[n];
	for(i=0;i<n;i++)cin>>p[i];
	double dp[n+1][n+1],ans=0.0;
	dp[1][0]=1.0-p[0];
	dp[1][1]=p[0];
	for(i=2;i<=n;i++)
	{
		for(j=0;j<=i;j++)
		{
			dp[i][j]=dp[i-1][j]*(1.0-p[i-1]);
			if(j>0)dp[i][j]+=dp[i-1][j-1]*p[i-1];
		}
	}
	for(i=n/2+1;i<=n;i++)ans+=dp[n][i];
	printf("%0.15lf\n",ans);
	return 0;
	
}